Numbers – Help With Factors and Multiples

By Stuart Ackerman

     In order to understand factors and multiples, students must have a good understanding of multiplication and division.

     Let’s take a look at the following equation,

3 x 5 = 15

     The number 15 is a multiple of 5 because it can be evenly divided by 3.  The number 15 is also a multiple of 3 because it can be evenly divided by 5.

3 x 5 = 15

3 and 5 are both factors of 15

15 is a multiple of both 3 and 5

     Factors are numbers that are multiplied to get a product (a product is the result of multiplication).  Finding a factor is like multiplying backwards.

     For example:

The factors of 12 are….. 1, 2, 3, 4, 6, 12

because all these numbers multiply evenly to get 12.  That is, 1 x 12 = 12, 2 x 6 = 12, and 3 x 4 = 12.

10 ….. 1, 2, 5, 10

20….. 1, 2, 4, 5, 10, 20

    Factors are either composite numbers or prime numbers. A prime number has only two factors, one and itself because it can’t be divided evenly by any other numbers. 

     For example, the numbers 7 and 11 are prime numbers.  The only numbers that divide evenly into 7 are 1 and 7.  Similarly, the only numbers that divide into 11 are 1 and 11.

     A composite number is any number that has more than two factors. 

     For example, 6 is a composite number because 1, 2, 3 and 6 can all divide into 6 evenly.

     We can write any composite number as a product of prime factors. This is called prime factorization. To find the prime factors of a number, you divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number.

For example, let’s try to find the factors for the number 48.  Since 48 is an even number, we can divide it by 2 (48 = 2 x 24).

48

2 x 24

We can see that the number 2 has no factors except for 2 and 1, so we will find the factors for 24.  Again, since it’s even, we’ll divide by 2 (2 x 12 = 24).

48

2 x 24

2 x 2 x 12

The 2’s cannot be factored any further, so, we will factor the 12 (2 x 6 = 12).

48

2 x 24

2 x 2 x 12

2 x 2 x 2 x 6

Again, the 2’s cannot be factored any further so we will factor the 6 (2 x 3 = 6).

48

2 x 24

2 x 2 x 12

2 x 2 x 2 x 6

2 x 2 x 2 x 2 x 3

The prime factors of 48 are 2 x 2 x 2 x 2 x 3.  We can check our work by actually multiplying the factors.

2 x 2 = 4 x2 = 8 x 2 = 16 x 3 = 48.

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